Average Error: 58.2 → 0.6
Time: 30.7s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]
↓
\[\left(\mathsf{fma}\left(\frac{-1}{60}, \left(\left(im \cdot im\right) \cdot im\right) \cdot \left(im \cdot im\right), im \cdot -2\right) \cdot \cos re\right) \cdot 0.5 + \left(\left(\left(im \cdot im\right) \cdot im\right) \cdot \frac{-1}{3}\right) \cdot \left(\cos re \cdot 0.5\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)
↓
\left(\mathsf{fma}\left(\frac{-1}{60}, \left(\left(im \cdot im\right) \cdot im\right) \cdot \left(im \cdot im\right), im \cdot -2\right) \cdot \cos re\right) \cdot 0.5 + \left(\left(\left(im \cdot im\right) \cdot im\right) \cdot \frac{-1}{3}\right) \cdot \left(\cos re \cdot 0.5\right)
double f(double re, double im) {
        double r6311743 = 0.5;
        double r6311744 = re;
        double r6311745 = cos(r6311744);
        double r6311746 = r6311743 * r6311745;
        double r6311747 = 0.0;
        double r6311748 = im;
        double r6311749 = r6311747 - r6311748;
        double r6311750 = exp(r6311749);
        double r6311751 = exp(r6311748);
        double r6311752 = r6311750 - r6311751;
        double r6311753 = r6311746 * r6311752;
        return r6311753;
}
↓
double f(double re, double im) {
        double r6311754 = -0.016666666666666666;
        double r6311755 = im;
        double r6311756 = r6311755 * r6311755;
        double r6311757 = r6311756 * r6311755;
        double r6311758 = r6311757 * r6311756;
        double r6311759 = -2.0;
        double r6311760 = r6311755 * r6311759;
        double r6311761 = fma(r6311754, r6311758, r6311760);
        double r6311762 = re;
        double r6311763 = cos(r6311762);
        double r6311764 = r6311761 * r6311763;
        double r6311765 = 0.5;
        double r6311766 = r6311764 * r6311765;
        double r6311767 = -0.3333333333333333;
        double r6311768 = r6311757 * r6311767;
        double r6311769 = r6311763 * r6311765;
        double r6311770 = r6311768 * r6311769;
        double r6311771 = r6311766 + r6311770;
        return r6311771;
}

Error

Bits error versus re

Bits error versus im

Target

Original58.2
Target0.2
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 58.2

    \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]

  2. Taylor expanded around 0 0.6

    \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\frac{1}{3} \cdot {im}^{3} + \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)\right)}\]

  3. Simplified0.6

    \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{3}, im \cdot \left(im \cdot im\right), \mathsf{fma}\left(im, -2, \frac{-1}{60} \cdot {im}^{5}\right)\right)}\]
  4. Using strategy rm

  5. Applied fma-udef0.6

    \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(\frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right) + \mathsf{fma}\left(im, -2, \frac{-1}{60} \cdot {im}^{5}\right)\right)}\]

  6. Applied distribute-rgt-in0.6

    \[\leadsto \color{blue}{\left(\frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot \left(0.5 \cdot \cos re\right) + \mathsf{fma}\left(im, -2, \frac{-1}{60} \cdot {im}^{5}\right) \cdot \left(0.5 \cdot \cos re\right)}\]

  7. Simplified0.6

    \[\leadsto \left(\frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right)\right) \cdot \left(0.5 \cdot \cos re\right) + \color{blue}{0.5 \cdot \left(\mathsf{fma}\left(\frac{-1}{60}, \left(im \cdot im\right) \cdot \left(\left(im \cdot im\right) \cdot im\right), -2 \cdot im\right) \cdot \cos re\right)}\]

  8. Final simplification0.6

    \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{60}, \left(\left(im \cdot im\right) \cdot im\right) \cdot \left(im \cdot im\right), im \cdot -2\right) \cdot \cos re\right) \cdot 0.5 + \left(\left(\left(im \cdot im\right) \cdot im\right) \cdot \frac{-1}{3}\right) \cdot \left(\cos re \cdot 0.5\right)\]

Reproduce

herbie shell --seed 2019146 +o rules:numerics
(FPCore (re im)
  :name "math.sin on complex, imaginary part"

  :herbie-target
  (if (< (fabs im) 1) (- (* (cos re) (+ (+ im (* (* (* 1/6 im) im) im)) (* (* (* (* (* 1/120 im) im) im) im) im)))) (* (* 0.5 (cos re)) (- (exp (- 0 im)) (exp im))))

  (* (* 0.5 (cos re)) (- (exp (- 0 im)) (exp im))))